Probabilistic Adaptive Width of a Multivariate Sobolev Space Equipped with a Gaussian Measure

In this paper, we determine the asymptotic values of the probabilistic adaptive widths of the space of multivariate functions with bounded mixed derivative (MW₂^r(T^d),μ) relative to the manifold (Y^N,ν) in the L_q(T^d)-norm, 1 < q ≤ 2, where μ and ν are two given Gaussian measures.     

Probabilistic and Average Linear Widths of Sobolev Space with Gaussian Measure in L \infty-Norm

In this paper we investigate the probabilistic linear $(n,\delta)$-widths and $p$-average linear $n$-widths of the Sobolev space $W^r_2$ equipped with the Gaussian measure $\mu$ in the $L_{\infty}$-norm, and determine the asymptotic equalities \begin{eqnarray*} \lambda_{n,\delta}(W^r_2,\mu,L_{\infty}) &\asymp&\frac{\sqrt{\ln (n/\delta)}}{n^{r+(s-1)/2}},\\[3pt] \lambda^{(a)}_n(W^r_2,\mu,L_{\infty})_p &\asymp&\frac{\sqrt{\ln n}}{n^{r+(s-1)/2}}, \qquad 0 < p < \infty. \end{eqnarray*}     

Approximation of functions on the Sobolev space with a Gaussian measure

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Valle-Poussin operators, Ces`aro operators, Abel opera-tors, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the L q space for 1≤q ∞, and the Fourier partial summation operators and the Valle-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the L q space for 1≤q ∞.     

Stochastic differential equations with coefficients in Sobolev spaces

We consider the Itô stochastic differential equation on Rd. The diffusion coefficients A₁,…,Am are supposed to be in the Sobolev space with p>d, and to have linear growth. For the drift coefficient A₀, we distinguish two cases: (i) A₀ is a continuous vector field whose distributional divergence δ(A₀) with respect to the Gaussian measure γd exists, (ii) A₀ has Sobolev regularity for some p^′>1. Assume for some λ₀>0. In case (i), if the pathwise uniqueness of solutions holds, then the push-forward #(Xt)γd admits a density with respect to γd. In particular, if the coefficients are bounded Lipschitz continuous, then Xt leaves the Lebesgue measure Lebd quasi-invariant. In case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish existence and uniqueness of stochastic flow of maps.     

N-Widths and Average Widths of Besov Classes in Sobolev Spaces

In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively.     

Second derivatives of convex functions in the sense of A. D. Aleksandrov on infinite-dimensional spaces with measure

We consider convex functions on infinite-dimensional spaces equipped with measures. Our main results give some estimates of the first and second derivatives of a convex function, where second derivatives are considered from two different points of view: as point functions and as measures.     

The Luzin approximation of functions from classes Wαp on metric spaces with measure

In this paper we prove an analog of the Luzin theorem on correction for spaces of the Sobolev type on an arbitrary metric space with a measure, satisfying the doubling condition. The correcting function belongs to the H?lder class and approximates a given function in the metrics of the initial space. Dimensions of exceptional sets are evaluated in terms of Hausdorff capacities and volumes. Original Russian Text ? V.G. Krotov and M.A. Prokhorovich, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 55–66. Dedicated to the memory of Petr Lavrent’evich Ul’yanov     

Optimal recovery on the classes of functions with bounded mixed derivative

Temlyakov considered the optimal recovery on the classes of functions with bounded mixed derivative in the Lp metrics and gave the upper estimates of the optimal recovery errors. In this paper, we determine the asymptotic orders of the optimal recovery in Sobolev spaces by standard information, i.e., function values, and give the nearly optimal algorithms which attain the asymptotic orders of the optimal recovery.     

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

     

Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness

This paper is a continuation of work of the author and joint work with Winfried Sickel. Here we shall investigate the asymptotic behaviour of Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness into Lebesgue spaces.     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

On behavior at infinity of the solutions to equations with dominating mixed derivative

We prove a theorem on the polynomial asymptotics at infinity for the solutions to differential equations with dominating mixed derivative with constant coefficients.     

最坏框架与平均框架下区间[1,1]上带Jacobi权的函数逼近

本文研究最坏框架和平均框架下区间[1,1]上带Jocobi权(1 x)α(1+x)β,α,β1/2的函数逼近问题.在最坏框架下,本文得到加权Sobolev空间BWr p,α,β在Lq,α,β(1 q∞)空间尺度下的Kolmogorov n-宽度和线性n-宽度的渐近最优阶,其中Lq,α,β(1 q∞)表示区间[1,1]上带Jacobi权的加权Lq空间.在平均框架下,本文研究具有Gauss测度的加权Sobolev空间Wr2,α,β被多项式子空间和Fourier部分和算子在Lq,α,β(1 q∞)空间尺度下的最佳逼近问题,得到平均误差估计的渐近阶.我们发现,在平均框架下,多项式子空间和Fourier部分和算子在Lq,α,β(1 q2+22 max{α,β}+1)空间尺度下是渐近最优的线性子空间和渐近最优的线性算子.     

Embedding theorems in the Sobolev-Morrey type spaces S p,a,ℵτ W W(G) with dominant mixed derivatives

We construct Sobolev-Morrey type spaces with dominant mixed derivatives and, using the obtained integral representation, prove some embedding theorems in these spaces.